Hidden Markov Tree Model of the Uniform Discrete Curvelet Transform Image for DenoisingIntroductionImplementationUDCT Marginal StatisticsConditional Distribution (1)Conditional Distribution (2)Hidden Markov Tree (HMT) ModelTree Structure of UDCTHMT (1)HMT (2)Denoising (1)Denoising (2)Denoising (3)Denoising Results (1)Denoising Results (2)Denoising Results (3)Denoising Results (4)Denoising Results (5)Hidden Markov Tree Model of the Uniform Discrete Curvelet Transform Image for DenoisingYothin RakvongthaiIntroduction•Curvelet Transform (Candes&Donoho 1999)•Implementation: –Fast Discrete Curvelet Transform (FDCT) (Candes et. al 2005) : in frequency domain–Contourlet (Do&Vetterli 2005) : in time domain with wavelet-like tree structure•Uniform Discrete Curvelet Transform (UDCT) (Nguyen&Chauris 2008) : in frequency domain with wavelet-like tree structureImplementationUDCT Marginal StatisticsKurtosis = 24.42Kurtosis = 23.71Kurtosis = E[(x-μ)4]/σ4 . Kurtosis of Gaussian = 3Conditional Distribution (1)•On parent (same position in next level)P(X|PX)Bow-tie shape uncorrelated but dependentConditional Distribution (2)•On parent•P(X|PX=px)•Kurtosis=3.51•~GaussianHidden Markov Tree (HMT) Model•Conditional distribution is Gaussian•X depends on PX Use HMT to model the coefficients•HMT model links between the hidden state variables of parent and children•HMT parameters (parameters of the density function) can be trained using the expectation-minimization (EM) algorithmTree Structure of UDCTHMT (1)•c(j,k,n) – coefficient in scale j, direction k, position n•S(j,k,n) – hidden state taking on values m = “S” or “L” with density function P(S(j,k,n))•Conditioned on S(j,k,n)=m, c(j,k,n) is Gaussian with mean μm(j,k,n) and variance σ2m(j,k,n) (m=Ssmall variance, m=Llarge variance)HMT (2)•The total pdf•P(S(j,k,n)), μm(j,k,n), σ2m(j,k,n) can be trained from the EM algorithm (Crouse et al 1998).•Define Θ = set of P(S(j,k,n)), μm(j,k,n), σ2m(j,k,n)Denoising (1)Problem formulation: y = x+w–ynoisy coefficients–xdenoised coefficients–wnoise coefficients with known varianceWant to estimate x from the knowledge of y and variance of wDenoising (2)•Obtain Θ from EM algorithm•The variance of denoised coefficients isDenoising (3)•The estimate of xDenoising Results (1)PSNR = Peak Signal to Noise RatioDenoising Results (2)SSIM = Structure Similarity Index (Wang et. al 2004)Denoising Results (3)Contourlet (25.85dB) DT-CWT (26.54dB) UDCT (27.32dB)Original Noisy (14.14dB) Wavelet (25.73dB) (SSIM 0.112) (SSIM 0.561)(SSIM 0.590) (SSIM 0.579) (SSIM 0.676)Denoising Results (4) Original Noisy (14.14dB) Wavelet (23.38dB) Contourlet (22.94dB) DT-CWT (24.15dB) UDCT (24.35dB) (SSIM 0.184) (SSIM 0.508) (SSIM 0.479) (SSIM 0.557) (SSIM 0.570)Denoising Results (5) Original Noisy (14.14dB) Wavelet (25.25dB) Contourlet (25.51dB) DT-CWT (25.99dB) UDCT (26.51dB) (SSIM 0.110) (SSIM 0.539) (SSIM 0.555) (SSIM 0.553) (SSIM
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